Group completion is a completion

D e p a r t m e n t o f M a t h e m a t i c a l S c i e n c e sU N I V E R S I T Y O F C O P E N H A G E N

January 25, 2020

Group completion is acompletionOscar Bendix Harr

Advisor: Jesper Grodal

Contents

Abstract 3

Acknowledgements 4

0 Introduction 50.1 A brief history of group completion . . . . . . . . . . . . . . . 50.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60.3 Notation and terminology . . . . . . . . . . . . . . . . . . . . 6

1 Localization 71.1 Localization of ordinary categories . . . . . . . . . . . . . . . . 71.2 Localization of ∞-categories . . . . . . . . . . . . . . . . . . . 141.3 Ore localization of ring spectra . . . . . . . . . . . . . . . . . 22

2 Group completion 272.1 Delooping E1-groups . . . . . . . . . . . . . . . . . . . . . . . 282.2 The group completion theorem . . . . . . . . . . . . . . . . . 302.3 The (−)∞ construction . . . . . . . . . . . . . . . . . . . . . . 312.4 Connection with the plus construction . . . . . . . . . . . . . 352.5 The Barratt-Priddy-Quillen theorem . . . . . . . . . . . . . . 36

A Appendix 39

Abstract

Classically, group completion relates the singular homology of atopological monoid M to the singular homology of ΩBM [BP72, Qui94,MS76]. Following Nikolaus [Nik17], we present a higher categoricalproof of the group completion theorem for E1-monoids and general-ized homology theories.

The technical ingredient in this proof is the theory of Bousfield lo-calizations in the setting of stable and locally presentable∞-categories,in particular the special case of Ore localization of E1-ring spectra. Wegive an introduction to this theory after reviewing the correspondingtheory for ordinary categories.

The higher categorical approach allows a simple proof of a well-known (but only recently proved in full [RW13]) theorem which con-nects group completion to Quillen’s plus construction. In particular,this theorem yields the striking result Ω∞S ' Z×BΣ+

∞.

Resume

Klassisk gruppefuldstændiggørelse relaterer den singulære homologiaf en topologisk monoid M til den singulære homologi af ΩBM [BP72,Qui94, MS76]. Vi præsenterer et bevis fra Nikolaus [Nik17] for grup-pefuldstændiggørelsessætningen for E1-monoider og generaliserede ho-mologiteorier.

Den tekniske ingrediens i dette bevis er teorien om Bousfieldlokalis-ering af stabile og lokalt præsentable ∞-kategorier, især specialtil-fældet af Orelokalisering af E1-ringspektra. Vi introducerer denneteori efter at have gennemgaet den tilsvarende teori for almindeligekategorier.

Den højere kategoriske tilgang tillader et simpelt bevis for en velk-endt (men først for nylig bevist til fulde [RW13]) sætning, der forbindergruppefuldstændiggørelse til Quillens pluskonstruktion. Specielt medførerdenne sætning det slaende resultat Ω∞S ' Z×BΣ+

∞.

Acknowledgements

I wish to extend the warmest thanks to Jesper Grodal, both for his insightfulguidance as my supervisor on this project and for teaching me homotopytheory. In my meetings with him and through his lectures, he repeatedlyrefreshes my enthusiasm for the subject.

I am also deeply grateful to Lars Hesselholt for recommending Nikolaus’sarticle and for many illuminating discussions about higher category theoryand higher algebra. I am thankful for his valuable comments on an earlierversion of § 1.

I am indebted to my dear friend Max Fischer, whose thorough reading ofmy drafts saved me from embarrassing mistakes and ambiguities.

I would also like to thank participants in the Topics in Algebraic Topologyseminar course for helpful comments on my presentation of Nikolaus’s article.Finally, I wish to thank my friends at the institute who, throughout thewriting process, accompanied me on an unhealthy number of trips to thecoffee machine in the break room.

0 INTRODUCTION

0 Introduction

In algebraic topology, every loop space has a canonical multiplication mapwhich is associative up to (coherent) homotopy: loop concatenation. Givenan E1-monoid, meaning a space with a multiplication map which is associa-tive up to coherent homotopy, it is therefore natural to ask whether the spaceis actually a loop space.

Loop concatenation always comes with inverses. It follows that in orderfor an E1-monoid to be a loop space, it must have inverses. A classicaltheorem of Stasheff says that failing to have inverses is the only obstructionto delooping. In modern terms, any E1-group is a loop space.

The delooping of E1-groups is functorial, and the functor is defined moregenerally for E1-monoids. This functor is denoted by B since it extends thewell-known classifying space functor G 7→ BG, originally defined for topolog-ical groups. Hence every E1-monoid M has an associated E1-group ΩBM ,and M ' ΩBM if and only if M is an E1-group. The group completiontheorem describes the relationship between M and ΩBM in general.

0.1 A brief history of group completion

The first group completion theorem appeared in [BP72]. In the introductionto that paper, Barratt and Priddy write that they discovered the theorem intheir efforts to understand (what is now called) the Barratt-Priddy-Quillentheorem (see § 2.5 below). Quillen gave another early proof of the theorem,finally published as an appendix in [Qui94]. Both proofs are based on spectralsequence calculations.

McDuff and Segal gave a more conceptual proof in [MS76], which re-mains a standard reference. The idea of their proof is to exhibit a homologyfiber sequence X → XM → BM for each space X on which M (a topo-logical monoid) acts via homology equivalences. When X is taken to be acertain well-understood space M∞, one finds that the middle term of thissequence is contractible, hence M∞ is homology equivalent to ΩBM . Thisapproach is inspired by Segal’s proof of the delooping theorem for E1-groupsin [Seg74] (see § 2.1 below). More than a decade later, Moerdijk published aproof [Moe89] based on the McDuff-Segal approach which avoids the use ofspectral sequences completely, making for a pleasantly abstract proof.

Here we follow the approach in Nikolaus’s exposition article [Nik17]. Theproof given there has several advantages compared to older proofs. For one, it

5

0.2 Prerequisites 0 INTRODUCTION

justifies the use of the term “group completion” by showing that the functorM 7→ ΩBM is a completion (or localization) in Bousfield’s sense, whencethe title of this project. This insight allows elegant proofs of both the maingroup completion theorem in its proper setting of E1-monoids (rather thanstrict topological monoids!) and the theorem conneting group completion tothe plus construction, which was not proved in full before [RW13].

0.2 Prerequisites

This exposition is aimed at advanced students with a background in homo-topy theory. Specifically, we assume familiarity with simplicial sets, homo-topy (co)limits and (ring) spectra.

We use language and results from higher category theory and higher al-gebra. One goal of this exposition is to demonstrate the power of this frame-work by proving a classical (and historically quite difficult) result using themachinery of Bousfield localization of∞-categories (following [Nik17]). Am-ple textbook references are given, especially to Lurie’s books: Higher ToposTheory (HTT) [Lur09] and Higher Algebra (HA) [Lur17]. We believe thatreaders who are unfamiliar with higher nonsense can still appreciate the ideasthat underpin this project, all of which are classically motivated. In partic-ular, we urge such readers to think of (co)limits in ∞-categories (e.g. the∞-category of spaces) as homotopy (co)limits (HTT 4.2.4.1).

0.3 Notation and terminology

• We fix a base Grothendieck universe whose elements are referred to assmall sets. The category of small sets is denoted by Sets.

• Following Lurie, we always think of ∞-categories simply as quasicate-gories, meaning simplicial sets which have the right extension propertywith respect to inner horn inclusions Λn

i ⊆ ∆n (0 < i < n).

• The ∞-category of spaces, denoted S, is the homotopy coherent nerve(as in HTT 1.1.5) of the category of Kan complexes enriched over Kancomplexes via the standard mapping complex construction.

• In keeping with older conventions (but breaking with Lurie), we use theterm locally presentable category (resp. ∞-category) instead of simply“presentable category” (resp. “∞-category”).

6

1 LOCALIZATION

1 Localization

In this section, we set up the localization machinery that will be needed in § 2.In § 1.1, we present the theory of Bousfield localization for locally presentablecategories. This motivates the corresponding theory for ∞-categories whichwe study in § 1.2. In both settings, the proofs of the main theorems arebased on small object arguments, and are therefore elegant and conceptual.Finally, we specialize to the case of Ore localization of ring spectra in § 1.3.

1.1 Localization of ordinary categories

Fix a category C and a set of morphisms W in C. Recall that a localizationof C at W is a functor q : C → C[W−1] which (1) sends every morphism inW to an isomorphism and (2) is initial among functors satisfying (1) in theenriched sense, meaning that

Fun(C[W−1],D)q∗−→ Fun∼W (C,D)

is an equivalence of categories for all categories D, where Fun∼W (C,D) de-notes the full subcategory of Fun(C,D) spanned by functors that invert mor-phisms in W .

In general the localization C[W−1] does not have a nice description. Evenwhen C is locally small and W is a small set, we cannot expect the localizationto be locally small.

On the other hand, many localizations which occur in nature are well-behaved and admit a description in terms of so-called local objects which liveinside the original category. This idea is due to Adams, but the set-theoreticissues were overcome by Bousfield [Bou75].

1.1 Definition. An object c ∈ C is W -local if the induced map

homC(y, c)f∗−→ homC(x, c)

is a bijection for all f : x → y in W . The full subcategory of C spanned byW -local objects is denoted by CW .

1.2 Definition. A morphism f : x→ y is a W -equivalence if

homC(y, c)f∗−→ homC(x, c)

is a bijection for each c ∈ CW . The set of W -equivalences is denoted by W .

7

1.1 Localization of ordinary categories 1 LOCALIZATION

Intuitively, a W -equivalence is a map which looks like an isomorphism tothe W -local objects. Indeed:

1.3 Proposition (Whitehead’s theorem). Suppose f : d→ d′ is a W -equivalencewhere d and d′ are W -local. Then f is an isomorphism.

Proof. By assumption homC(d′, c)f∗−→ homC(d, c) is an bijection for all c ∈

CW . But here d and d′ are also in CW , and since the latter is a full subcategoryof C, the hom-sets identify with hom-sets in CW . The conclusion follows fromYoneda’s lemma.

Clearly every element of W is a W -equivalence. The opposite inclusionis false in general. Due to the following proposition, it is important to un-derstand the relationship between W and W .

1.4 Proposition. Suppose L : C→ CW is left adjoint to the inclusion CW ⊆C. Then

(1) A map f in C is a W -equivalence if and only if Lf is an isomorphism.

(2) The map L : C→ CW exhibits CW as the localization of C at W .

Proof. For f : x→ y and any c ∈ CW , the adjunction supplies a commutativediagram:

homCW(Ly, c) homCW

(Lx, c)

homC(y, c) homC(x, c)

(Lf)∗

∼ ∼

f∗

By definition, the bottom arrow is a bijection for all c ∈ CW if and only if fis a W -equivalence. By Yoneda’s lemma, the top arrow is a bijection for allc ∈ CW if and only if Lf is an equivalence.

We prove (2). Let u be the unit of the adjunction. We claim that ux : x→Lx is a W -equivalence for all x ∈ C. Since u is the unit of the adjunction,we have that

homCW(Lx, c)→ homC(Lx, c)

u∗x−→ homC(x, c)

is a bijection for all x ∈ C and c ∈ CW . Since CW is a full subcategory, thefirst map is already a bijection, showing the claim.

8


Let D be any category. We will show that the functor Fun(CW ,D)L∗−→

Fun∼W (C,D) is fully faithful and essentially surjective.To see that L∗ is essentially surjective, suppose F : C → D is a functor

which inverts W . Denoting the inclusion CW ⊆ C by i, we claim that F isnaturally isomorphic to L∗(Fi) = FiL. Indeed, Fu : FiL → F is a natural

isomorphism since F inverts every component of u, all of which lie in W bywhat we have just seen.

To see that L∗ is fully faithful, we may assume that the composition Liis the identity on CW .1 If η : FL → GL is a natural transformation whereF and G are functors from CW to D, it follows that η = L∗(ηi). This showsthat every natural transformation from F to G lies in the image of L∗. Theproof of injectivity is similar.

We also have a helpful criterion for the existence of such a left adjoint.

1.5 Definition. Let x ∈ C. A map f : x→ c is a W -localization of x if f isa W -equivalence and c is W -local.

1.6 Proposition. The inclusion CW ⊆ C has a left adjoint if and only ifeach x ∈ C has a W -localization u : x→ c.

Proof. Assume first that the inclusion has a left adjoint L and let u be theunit of the adjunction. We saw in the proof of the Proposition 1.4 thateach component ux : x → Lx is a W -equivalence, and by assumption Lx isW -local, so ux is a W -localization of x.

Now assume that there is a W -equivalence ux : x → cx for each x ∈ C.We wish to define a functor L : C → CW by Lx = cx on objects such thatthe collection ux : x→ Lx forms the unit of an adjunction between L (onthe left) and the inclusion CW ⊆ C (on the right). We must define L onmorphisms. But since Ly is W -local and ux is a W -equivalence, we have

homC(Lx, Ly)u∗x−→ homC(x, Ly),

so for each f : x→ y there is a unique dashed arrow such that

x y

Lx Ly

ux

f

uy

Lf

1This follows from the proof of Proposition 1.6.

9


commutes. Uniqueness implies that L respects composition and takes iden-tity morphisms to identity morphisms. Hence L is a functor and by con-struction ux is a natural transformation from the identity functor on C

to the composition of L and the inclusion of CW back into C. Furthermore,it follows from the argument at the start of this proof (but in the oppositedirection) that ux forms the unit of an adjunction between L (on the left)and the inclusion CW ⊆ C (on the right).

From now on, assume for simplicity that C contains small colimits. Recallthat no assumptions were placed on the set W . On the other hand, we willsee that W has certain restrictive properties. The main theorem of thissubsection shows that W is generated by W under these properties as longas we make some mild assumptions on C and W .

1.7 Definition. A set of morphisms S in C is weakly saturated if

(1) S contains all isomorphisms.

(2) (Closure under cobase-change) Given a pushout

x x′

y y′

f f ′

in C, if f ∈ S then f ′ ∈ S.

(3) (Closure under transfinite composition) If λ is an ordinal and x : λ→ C

is a functor such that for every α ∈ λ with α 6= 0, the map

lim−→β<α

x(β)→ x(α)

is in S, then the canonical map x(0)→ lim−→β<λx(β) is in S.

(4) (Closure under retracts) Given a commutative diagram

a x a

b y y

f g f

where both horizontal compositions are identity morphisms, if g ∈ Sthen f ∈ S.

10


1.8 Definition. A weakly saturated set S is saturated if it has the followingadditional property:

(5) (Two-out-of-three) Given a commutative diagram

x y

z

if two of the arrows lie in S, then the third arrow also lies in S.

Remark. Note that an intersection of (weakly) saturated sets is again (weakly)saturated.

The set of isomorphisms in C is saturated. Clearly it is then the smallestsaturated set. More generally, if D is another category which contains smallcolimits and F : C→ D is a functor which preserves small colimits, then theset of morphisms in C which are sent to isomorphisms in D is a saturatedset.

1.9 Proposition. The set of W -equivalences is saturated.

Proof. For each z ∈ C, we view Fz = homC(−, z) as a functor from C toSetsop. Note that Fz preserves colimits. Hence

Sz = f : x→ y | homC(y, z)f∗−→ homC(y, z) is an iso

is a saturated set for each z ∈ C. It follows that the intersection⋂c∈CW

Sc = f : x→ y | homC(y, c)f∗−→ homC(y, c) is an iso for all c ∈ CW

is a saturated set. But this intersection is precisely W .

Since intersections of saturated sets are again saturated, the followingdefinition makes sense:

1.10 Definition. Let S be any set of morphisms in C. The saturation ofS is the smallest saturated set containing S, denoted by sat(S). Similarly,the weak saturation of S is the smallest weakly saturated set containing S,denoted by satw(S).

Remark. Clearly satw(S) ⊆ sat(S).

We can now state the main theorem of this subsection:

11


1.11 Theorem (Bousfield localization). Suppose that the category C islocally presentable and contains a terminal object. Assume also that theset of morphisms W is small. Then

(1) For each x ∈ C, there is a map f : x → c with f ∈ sat(W ) andc ∈ CW .

(2) The inclusion CW ⊆ C has a left adjoint.

(3) The set of W -equivalences W equals the saturation sat(W ).

The proof uses the small object argument, which in its original form is dueto Quillen. We refer the reader to HTT A.1.2.5 for a proof of the versionused here.

1.12 Theorem (Small object argument). Suppose C is locally presentableand that S is a small set of morphisms in C. Then any morphism f : x→ yin C has a factorization

xλ−→ z

ρ−→ y

where λ lies in the weak saturation satw(S) and ρ has the right lifting propertywith respect to S.

Proof of Theorem 1.11. We first prove (1). For each f : x→ y in C, the foldmap of f is the dashed arrow coming from the universal property applied to

x y

y y ∪x y

y

f

f pushout id

id

Denote the fold map of f by ∇f .Put

W ′ = W ∪ ∇f | f ∈ W.This is again a small set, so Theorem 1.12 applies. Letting ∗ denote a terminalobject of C, this implies in particular that for each x ∈ C, the map x → ∗factors as

xλ−→ c→ ∗

12


where λ is in the saturation of W ′ and c → ∗ has the right lifting propertywith respect to W ′. Note that ∇f ∈ sat(W ) for each f ∈ W by conditions(1), (2) and (5) for saturated sets. Hence the saturation of W ′ is just thesaturation of W , so λ ∈ sat(W ).

We claim that c is W -local. Note that by definition c is W -local if andonly if for every solid arrow diagram

x c

y

f

with f ∈ W , there exists a unique dashed arrow making the diagram com-mute. This extension problem is equivalent to the lifting problem:

x c

y ∗f

We know that c→ ∗ has the right lifting property with respect to W ⊆ W ′, socertainly we can always find a dashed arrow solution to this lifting problem.We claim that such solutions are unique. Suppose g and h both solve thelifting problem for the same map x→ c. Then they fit into the diagram:

x y

y y ∪x y

c

f

f pushout g

h

Let g ∪ h denote the dashed arrow coming from the universal property asshown in the diagram. Since c→ ∗ has the right lifting property with respectto fold maps, we can solve the following lifting problem:

y ∪x y c

y ∗∇f

g∪h

13

1.2 Localization of ∞-categories 1 LOCALIZATION

But this implies that g = h, showing uniqueness.We claim that (2) follows from what we have just proved. Indeed, we

know that W is saturated and that W ⊆ W , so it follows that sat(W ) ⊆ W .Hence the existence of morphisms f : x → c in sat(W ) where c is W -localimplies the existence of a left adjoint by Proposition 1.6.

Denote the left adjoint by L. Recall from the proof of Proposition 1.6 thatthe W -equivalences f : x → c, which in our case come from the saturationsat(W ), form the unit of the adjunction between L and the inclusion. Denotethe collection of these maps by ux. For each map f : x→ y in C, naturalitygives a commutative diagram

x y

Lx Ly

ux

f

uy

Lf

If f ∈ W , we know from Proposition 1.4 (1) that Lf is an isomorphism, so inparticular it lies in the saturated set sat(W ). But then the two-out-of-three

property implies that f ∈ sat(W ). This shows that W ⊆ sat(W ), whichproves (3) since we know from above that the opposite inclusion holds.

Remark. Typically, the set of morphisms W that one wishes to localize isalready saturated, but fails to be small. In that case, the goal is to find asmall set W0 ⊆ W whose saturation is all of W . If this is possible, W is saidto be of small generation.

1.2 Localization of ∞-categories

The machinery from the previous subsection carries over to∞-categories. Interms of definitions and theorems, this passage from ordinary categories to∞-categories is predictable: simply replace hom-sets with mapping spacesand bijections with equivalences. The proofs, however, are often different for∞-categories.

Fix an ∞-category C and a set of morphisms W in C.

1.13 Definition. An object c ∈ C is W -local if the induced map

MapC(y, c)f∗−→ MapC(x, c)

14


is an equivalence for all f : x → y in W . The full subcategory of C spannedby W -local objects is denoted by CW .

As in the previous subsection, the goal is to find a functor L : C → CWwhich is left adjoint to the inclusion CW ⊆ C in the ∞-categorical sense.

Adjunctions are covered in HTT 5.2, especially 5.2.2 and 5.2.3. We willuse a different definition than Lurie, which has the advantage of being rec-ognizable to students of classical category theory – without having to be“straightened” first. Lurie proves that this definition agrees with the un-straightened definition in HTT 5.2.2.8.

1.14 Definition. Given a pair of parallel functors

C Cf

g

between ∞-categories, we say that f is a left adjoint to g if there exists anatural transformation u : idC → g f such that for every pair of objectsc ∈ C and d ∈ D, the induced map on mapping spaces

MapD(f(c), d)g∗−→ MapC(gf(c), g(d))

u(c)∗−−−→ MapC(c, g(d))

is an equivalence. In this case, we also say that g is a right adjoint to f , andthat u is a unit transformation.

Roughly speaking, ∞-categorical left adjoints (resp. right adjoints) be-have as one would expect:

(a) Left adjoints preserve colimits and right adjoints preserve limits (HTT5.2.3.5);

(b) If it exists, a left resp. right adjoint to a functor is unique up tocontractible choice (HTT 5.2.6.2);

(c) A version of Freyd’s adjoint functor theorem holds, i.e. for locallypresentable ∞-categories, every functor which preserves small colimitsis a left adjoint, and every accessible functor which preserves limits isa right adjoint (HTT 5.5.2.9).

With prerequisites out of the way, descent gives us a recognition principlefor endofunctors which are left adjoint to the inclusion of their image:

15


1.15 Proposition (HTT 5.2.7.4). Let L : C→ C be a functor with essentialimage LC. Then the following are equivalent:

(1) When viewed as a functor from C to LC, the functor L is a left adjointto the inclusion LC ⊆ C.

(2) There exists a natural transformation u : ∆1 × C → C from idC to Lsuch that for all x ∈ C the morphisms L(u(x)), u(L(x)) : Lx→ LLx ofare equivalences in C.

Proof. As in the proof of Proposition 1.6, the key point is that in the sequence

MapLC(Lx, Ly)→ MapC(Lx, Ly)u(x)∗−−−→ MapC(x, Ly),

the first map is already an equivalence since LC is by definition a full sub-category of C. It follows that the composition is an equivalence if and only ifu(c)∗ is an equivalence, and so u is a unit transformation if and only if thisholds for each x ∈ C and each Ly ∈ LC.

Assume that u is a unit transformation, so by our discussion the map u(c)∗

is always an equivalence. If also x ∈ LC (so that we are always really in LC),then the Yoneda lemma implies that u(x) is an equivalence. In particular,this shows that u(L(x)) is an equivalence. Naturality implies that we havecommutative diagram

x Lx

Lx LLx

u(x)

u(x)

u(Lx)

L(u(x))

Since u(x)∗ is an equivalence (hence injective on π0), we see that L(u(x)) 'u(Lx), so in particular L(u(x)) is also an equivalence.

Assume now that u is some natural transformation satisfying the condi-tion in (2). We must show that

MapC(Lx, Ly)u(x)∗−−−→ MapC(x, Ly)

is an equivalence for all x ∈ C and all Ly ∈ LC, i.e. an isomorphism in thehomotopy category of spaces. By the Yoneda lemma, it suffices to show that

homH (K,MapC(Lx, Ly))[u(x)∗ ]∗−−−−−→ homH (K,MapC(x, Ly))

16


is a bijection for every space K. But using the exponential rule, this mapidentifies with

homhFun(K,C)(δ(Lx), δ(Ly))→ homhFun(K,C)(δx, δ(Ly)),

where δ : C→ Fun(K,C) is the functor which sends c to the constant functorat c. Since we can replace C with Fun(K,C), it therefore suffices to showthat

homhC(Lx, Ly)[u(x) ]∗−−−−→ homhC(x, Ly)

is a bijection for all x ∈ C and all Ly ∈ LC.We first check surjectivity. Let f : x→ Ly. By naturality of u, we have

x Ly

Lx LLy

u(x)

f

u(Ly)∼Lf

Here our assumption says that u(Lx) is invertible, so choosing an inverse(u(Lx))−1, we can write

f ' (u(Lx))−1 Lf u(x) = u(x)∗((u(Lx))−1 Lf

).

As for injectivity, note that for each g : Lx→ Ly we have

Lx Ly

LLx LLy

∼

u(Lx)

g

u(Ly)∼

Lg

It follows that

g ' (u(Ly))−1 Lg u(Lx)

' (u(Ly))−1 Lg L(u(x)) (L(u(x)))−1 u(Lx)

' (u(Ly))−1 L (g u(x)) (L(u(x)))−1 u(Lx)

Here everything except the middle factor of L (g u(x)) is invertible by as-sumption. It follows that g is uniquely determined by g u(x), hence thatu(x)∗ is injective.

17


As before, we are interested in morphisms which look like equivalences tothe W -local objects.

1.16 Definition. A morphism f : x→ y in C is a W -equivalence if

MapC(y, c)f∗−→ MapC(x, c)

is an equivalence for each c ∈ CW . The set of W -equivalences is denoted byW .

Here is the analog of the characterization of W -equivalences from Propo-sition 1.4:

1.17 Proposition. Suppose L : C→ CW is left adjoint to the inclusion CW ⊆C. Then a map f ∈ C is a W -equivalence if and only if Lf is an equivalence.

Proof. This follows from the adjunction as in the proof of Proposition 1.4.

There is also a direct analog of Proposition 1.6.

1.18 Definition. Let x ∈ C. A map f : x → c is a W -localization of x if fis a W -equivalence and c is W -local.

1.19 Proposition (HTT 5.2.7.8). The inclusion CW ⊆ C has a left adjointif and only if each x ∈ C has a W -localization u : x→ c.

Replacing mapping spaces with hom-sets and equivalences with bijections,the argument from the proof of Proposition 1.6 can be used to show that thecomponents of the unit transformation which accompanies a left adjoint tothe inclusion are W -equivalences.

As for the other direction, we cannot (at least naively) use the techniquefrom the proof of the classical counterpart to this proposition, since a functorbetween ∞-categories consists of more than its values on objects and mor-phisms, and a natural transformation between such functors is more than acollection of morphisms. Instead, we will use a trick from the unstraight-ened world to show that the unstraightened condition for the existence ofa left adjoint is satisfied. Readers who are unfamiliar with the theory of(un)straightening can skip the proof.

18


Proof of Proposition 1.19. Define D to be the full subcategory of ∆1 × C

spanned by objects of the form (x, i) with x ∈ CW if i = 1. The projectionp : D→ ∆1 is the cartesian fibration associated to the inclusion CW ⊆ C viaunstraightening.

We claim that a morphism f : x → c with c ∈ CW is a W -equivalence ifand only if the associated morphism f : (x, 0)→ (c, 1) in D is p-cocartesian.According to the dual of HTT 2.4.4.3, f : (x, 0) → (c, 1) is p-cocartesian ifand only if the diagram

MapD((c, 1), (y, i)) MapD((x, 0), (y, i))

Map∆1(1, i) Map∆1(0, i)

f∗

is homotopy cocartesian for all (y, i) ∈ D.If i = 0, then both mapping spaces on the left are empty and the bot-

tom righthand corner is contractible, so there is nothing to check. Therefore,assume i = 1. Then the spaces appearing in the bottom row are both con-tractible. Also, since i = 1, the y appearing in the top row always belongsto CW . The top horizontal map then identifies with

MapC(c, y)f∗−→ MapC(x, y),

and the diagram is homotopy cocartesian if and only if this map is an equiv-alence. By definition, this holds for all y ∈ CW if and only if f is a W -equivalence. Hence a W -equivalence x → c with c ∈ CW corresponds to acocartesian lift with source (x, 0) of the unique morphism in ∆1.

We have shown that there exists W -equivalences from every object of Cto a W -local object if and only if the cartesian fibration D → ∆1, which isthe unstraightening of the inclusion CW ⊆ C, is also a cocartesian fibration.But this is precisely the unstraightened condition for the inclusion to have aleft adjoint (HTT 5.2.2.1).

Remark. In the situation of the previous proposition, the proof of HTT 5.2.2.8actually shows that the morphisms f : x→ c form the unit of the adjunction,just as in the classical case. We will need this in the proof of Theorem 1.22.

Again the relationship between W and W is, modulo set-theoretic as-sumptions, that W is a kind of saturation of W . Lurie proves a general

19


theorem of this sort (HTT 5.5.4.15), but using a stronger notion of satura-tion than we have been using. We stick to the weaker notions of saturationand weak saturation introduced above, which are defined correspondingly for∞-categories. The argument from the proof of Proposition 1.9 carries overto show that W is saturated.

Again we will base our proof on the small object argument. We need aversion of this result for locally presentable ∞-categories.

1.20 Definition. A morphism g : x → y in C has the right lifting propertywith respect to a morphism f : c→ d if every square

c x

d y

f g

can be extended to a 3-simplex

c x

d y

f g

More generally, the morphism g has the right lifting property with respect toa set of morphisms S in C if it has the right lifting property with respect toevery element of S.

1.21 Theorem (Small object argument [Lur18, 12.4.2.1]). Suppose that the∞-category C is locally presentable and that S is a small set of morphismsin C. Every morphism f : x→ y in C has a factorization

xλ−→ z

ρ−→ y

where λ lies in the weak saturation satw(S) and ρ has the right lifting propertywith respect to S.

The right lifting property has an equivalent definition in terms of mappingspaces. Namely, g : x → y has the right lifting property with respect tof : c→ d if and only if the induced map

MapC/y(d, x)

f∗−→ MapC/y(c, x) (1.1)

20


is surjective on path-components for all maps d → y. This property is tooweak for our purposes because we have to produce a W -equivalence, meaninga map that induces equivalences on the relevant mapping spaces.

In the previous subsection, we adjoined fold maps to W to turn a sur-jective map into a bijection. The same trick works here. But we need theinduced map on mapping spaces to be a bijection on homotopy groups in alldegrees, not just on π0. In order to achieve this, we make some simplifyingassumptions that are satisfied in our applications.

Say that a set of morphisms S in a stable ∞-category is closed underupward shifts if f ∈ S implies Σf ∈ S. Then:

1.22 Theorem (Bousfield localization). Suppose that the ∞-category C

is locally presentable and stable. Assume also that the set of morphismsW is small and closed under upward shifts. Then

(1) For each x ∈ C, there is a map f : x → c with f ∈ sat(W ) andc ∈ CW .

(2) The inclusion CW ⊆ C has a left adjoint.

(3) The set of W -equivalences W equals the saturation sat(W ).

Proof. Form W ′ by adjoining fold maps to W . For each x ∈ C, the smallobject argument produces a factorization

xλ−→ c→ 0,

where 0 is the zero object of C, λ lies in the saturation sat(W ′) = sat(W ), andc → 0 has the right lifting property with respect to W ′. We claim that c isW -local. Since 0 is, in particular, a terminal object, the lifting problem (1.1)is equivalent to the extension problem

MapC(d, x)f∗−→ MapC(c, x)

for f : c → d in W ′. The fold map trick shows that this map is a bijectionon π0. Since W is closed under upward shifts, the maps

MapC(Σnd, x)(Σnf)∗−−−−→ MapC(Σnc, x)

21

1.3 Ore localization of ring spectra 1 LOCALIZATION

are also surjections on path components for all n ≥ 0. But these mapsidentify with

Ωn MapC(d, x)Ωn(f∗)−−−−→ Ωn MapC(c, x),

so we conclude that f induces bijections on higher homotopy groups as de-sired.

Now (2) and (3) follow from (1) as in the proof of Theorem 1.11.

1.3 Ore localization of ring spectra

A special case of the theory developed above is the localization of a ring spec-trum at a subset of its ring homotopy groups. This theory, which seeminglyis due to Lurie, is the subject of HA 7.2.3. Here we follow the introductionfrom the appendix in [Nik17].

Fix an E1-ring spectrum A and a subset S ⊆ π∗A. As in algebra, we wantto find a universal map i : A → S−1A having i∗(S) ⊆ π∗(A

′)×. As we willsoon see, such a universal map always exists, and under certain conditionswe can describe how the process of localization affects π∗A.

1.23 Definition. A left A-module spectrum M is called S-local if S actsinvertibly on π∗M . Let LMod∼SA ⊆ LModA denote the full subcategoryspanned by S-local modules.

1.24 Theorem. The inclusion LMod∼SA ⊆ LModA has a left adjoint L.Furthermore, the left adjoint is given by L = S−1

modA ⊗A −, where S−1modA is

a uniquely determined (A,A)-bimodule with a map A → S−1modA of (A,A)-

bimodules.

Proof. We claim that the S-local modules are precisely the WS-local modules(in the sense of the previous subsection), where

WS = Σ−nA Σ−nRs−−−−→ Σ−nA | n ∈ Z, s ∈ S.

Here Rs denotes multiplication by s from the right. By definition, M isWS-local if and only if

MapA(Σ−nA,M)Rs∗

−−→ MapA(Σ−nA,M)

is an equivalence for all n and s. Of course, here R∗s = (Ls)∗. In LModA,we can identify the mapping space MapA(−,−) with the infinite deloopingof the mapping spectrum mapA(−,−) (see [Gep19, 3.2]). As in algebra,mapA(A,−) ' id. But then

22


Ω∞mapA(Σ−nA,M) Ω∞mapA(Σ−nA,M)

Ω∞Σn mapA(A,M) Ω∞Σn mapA(A,M)

Ω∞ΣnM Ω∞ΣnM

∼

Ls∗

∼

∼

Σn(Ls∗)

∼

ΣnLs

Here the bottom arrow is an equivalence for all n if and only if Ls : M →Mis an equivalence, proving our claim that the S-local modules are exactlythe WS-local modules. The ∞-category LModA is locally presentable (HA4.2.3.7), so Theorem 1.22 implies that the inclusion LMod∼SA ⊆ LModA hasa left adjoint L.

Clearly the subcategory LMod∼SA ⊆ LModA is closed under colimits, sothe composition

LModAL−→ LMod∼SA → LModA,

preserves colimits. By abuse of notation, we denote this composition by L.But then HA 4.8.4.1 (or more specifically HA 7.1.2.4) implies that

L ' S−1modA⊗A −

for some uniquely determined (A,A)-bimodule S−1modA. It follows immediately

that L(A) ' L ' S−1modA⊗A A ' S−1

modA. The map A→ S−1modA is simply the

unit of the adjunction.

1.25 Corollary. The bimodule S−1modA has a unique A-algebra structure such

that the canonical map i : A→ S−1modA is a map of algebras.

With this structure,

(1) The map i is the localization of A at S as an A-algebra, i.e. for eachA-algebra B,

MapAlgE1 (A)(S−1modA,B)→ Map∼SAlgE1 (A)(A,B),

where Map∼SAlgE1 (A)(A,B) is the subspace of Mapinv(S)AlgE1 (A)(A,B) spanned

by maps which invert S.

(2) Suppose that A is an R-algebra for some E1-ring spectrum R. Then iis the localization of A at S as an R-algebra.

23


Proof. Since L = S−1mod⊗A− is a localization, it follows from Proposition 1.15

that the canonical maps

L(A) = S−1modA⇒ S−1

modA⊗A S−1modA = LL(A)

are homotopic equivalences, so S−1modA is idempotent in the monoidal ∞-

category of (A,A)-bimodules and it follows abstractly that S−1modA is an al-

gebra in this ∞-category, i.e. pick an inverse S−1modA⊗A S

−1modA→ S−1

modA asthe multiplication map (see e.g. HA 4.8.2.9).

The abstract arguments in the proof of Theorem 1.24 can be used to showthe existence of a localization of A at S as an A-algebra, which we denoteby A→ S−1

algA. We have a diagram:

S−1modA

A

S−1algA

ρλ

Universal properties give unique dashed arrows in both directions. Unique-ness implies that λ ρ ' id and ρ λ ' id. Hence S−1

modA ' S−1algA. The same

argument shows that S−1algA is the localization of A at S as an R-algebra.

From now on, we let S−1A denote the localization of A at S in whateversense is relevant to the context. According to the corollary, this does notlead to ambiguity.

Theorem 1.24 says that the localization of A at S always exists. However,we saw that this was a purely formal fact. We would like to understandthe process of localization more concretely. In particular, we would like tounderstand how localization influences homotopy groups.

The problem of understanding localization concretely also shows up inalgebra. To prepare for the main theorem of this subsection, we need a briefdigression to the classical situation. Here the analog of Theorem 1.24 is:

1.26 Proposition. Let S be a subset of a graded ring A∗. Let LModA∗denote the category of graded left A∗-modules, and let LMod∼SA∗ denote thefull subcategory of S-local modules, i.e. modules on which S acts invertibly.Then the inclusion LMod∼SA∗ ⊆ LModA∗ has a left adjoint.

24


Proof. As in the proof of Theorem 1.24, one shows that the S-local modulesare exactly the WS-local modules where

WS = A∗−·s−−→ A∗ | s ∈ S.

Then Theorem 1.11 implies the existence of the desired left adjoint.

Of course, in algebra we are used to imposing certain restrictions on S(maybe even on A∗, but this is unnecessary). Under such restrictions, it ispossible to describe the localization S−1A∗ as a ring of left fractions withdenominators in S. We would not like our restrictions to be too restrictive,however. The Ore condition strikes a good balance:

1.27 Definition. Let A∗ be a graded ring. A subset S ⊆ A∗ satisfies theleft Ore condition if

(1) S is multiplicatively closed and contains the identity.

(2) For all a ∈ A∗ and s ∈ S, there is a′ ∈ A∗ and s′ ∈ S so that as′ = sa′.

(3) If as = 0 where a ∈ A∗ and s ∈ S, then there is t ∈ S with ta = 0.

1.28 Proposition. Suppose S ⊆ A∗ satisfies the left Ore condition. Let M∗be a left A∗-module. Define a relation ∼ on M∗ × S

(m, s) ∼ (m′, s′) iff ∃ a, a′ ∈ A∗ : am = a′m′ and as = a′s′.

The relation ∼ is an equivalence relation. Furthermore, there is a uniqueA∗-module structure on the quotient S−1M∗ = (M∗ × S) / ∼ so that the mapM∗ → S−1M∗ given by a 7→ [(a, 1)] is a module homomorphism exhibitingS−1M∗ as the universal localization of M∗ at S.

1.29 Definition. A map f : M∗ →M ′∗ of graded left A∗-modules is an S-nil

equivalence if

(1) For each m′ ∈M ′∗ there is s ∈ S so that sm′ ∈ im(f).

(2) For each m ∈ ker(f) there is s ∈ S so that sm = 0.

The following algebraic lemma will come in handy later:

1.30 Lemma. Let S ⊆ A∗ be a multiplicatively closed subset containing theidentity.

25


(1) S satisfies the left Ore condition if and only if right multiplication

A∗−·s−−→ A∗ is an S-nil equivalence for each s ∈ S.

(2) Assume that S satisfies the left Ore condition. Let f : M∗ → M ′∗ be a

map of left A∗-modules. Then there is a commutative diagram

M ′∗

M∗ S−1M∗

f

∼

where M∗ → S−1M∗ is the universal localization if and only if f is anS-nil equivalence and M ′

∗ is S-local.

Proof. (1) follows directly from definitions. As for (2), it follows from theproof of Proposition 1.26 that the universal localization M∗ → S−1M∗ lies

in the weak saturation of WS = A∗−·s−−→ A∗. But WS is contained in

the class of S-nil equivalences, which is easily seen to be saturated. HenceM∗ → S−1M∗ must in particular be an S-nil equivalence. By the two-out-of-three property, the induced map S−1M∗ →M ′

∗ is an S-nil equivalence if andonly if f is an S-nil equivalence. Since S-nil equivalences between S-localmodules are just isomorphisms, we conclude that f is an S-nil equivalence ifand only if the induced map S−1M∗ →M ′

∗ is an isomorphism.

We now return to the world of higher algebra. Let A be an E1-ringspectrum again.

1.31 Definition. A map f : M → M ′ of left A-module spectra is an S-nilequivalence if the induced map of π∗(M) → π∗(M

′) of left π∗A-modules isan S-nil equivalence.

1.32 Lemma. The class of S-nil equivalences in LModA is saturated.

Proof. Obviously the class of S-nil equivalences has the two-out-of-threeproperty and contains all equivalences. Closure under retracts and pushoutsfollows from the long exact sequence of homotopy groups. Closure undertransfinite composition follows from the fact that the sphere spectrum iscompact.

26

2 GROUP COMPLETION

1.33 Theorem (Lurie). Suppose S ⊆ π∗(A) satisfies the left Ore condi-tion. For every left A-module M , there is a unique isomorphism

ϕ : S−1π∗(M)∼−→ π∗(S

−1M)

so thatπ∗(S

−1M)

π∗(M) S−1π∗(M)

fϕ ∼

commutes. Here the bottom arrow is the localization of π∗(M) at S as aπ∗(A)-module.

Proof. In the proof of Theorem 1.24 we saw that the localizations M →S−1M come from the Bousfield localization of LModA at the set

WS = Σ−nA Σ−nRs−−−−→ Σ−nA | n ∈ Z, s ∈ S.

Since S is assumed to satisfy the left Ore condition, Lemma 1.30(1) says thatWS ⊆ S-nil equivalences. Saturating both sides and using Lemma 1.32,we find sat(WS) ⊆ S-nil equivalences. But then Theorem 1.22 implies thatthe localization M → S−1M is an S-nil equivalence, and since π∗(S

−1M) isS-local, the result follows from Lemma 1.30(2).

2 Group completion

This section forms the heart of our exposition. We first prove a classicaldelooping theorem for E1-groups2 in § 2.1. Using this result, we prove thegroup completion theorem in § 2.2. The construction in § 2.3 allows us tounderstand the group completion theorem via a stable equivalence at thelevel of spaces, and in § 2.4 we see that the plus construction makes thisequivalence an actual equivalence. Finally, we describe a classical examplein § 2.5

2In older literature, these are called grouplike.

27

2.1 Delooping E1-groups 2 GROUP COMPLETION

2.1 Delooping E1-groups

Categorical products endow the category of Kan complexes enriched overKan complexes with a symmetric monoidal structure. This structure lifts toa symmetric monoidal structure on the ∞-category of spaces. Throughoutwhat follows, an E1-monoid (resp. E∞-monoid) will mean an E1-algebra(resp. E∞-algebra) in the symmetric monoidal∞-category of spaces. For thereader’s convenience, we have spelled out these definitions in more elementaryterms in the appendix.

Since the∞-category of spaces contains small colimits, the following def-inition makes sense:

2.1 Definition (HTT 6.1.2.12). Let A• be a simplicial space. The geometricrealization of A• is the colimit

‖A•‖ = lim−→N∆op

A•.

When A• = M• is an E1-monoid, we instead refer to the geometric realizationof M• as the classifying space of M• and, abusing notation, we write BM =‖M•‖.

Let M• be an E1-monoid. We have a commutative diagram

M M0

M0 BM

d0

d1

where M0 → BM is simply the canonical map into the colimit. Here M0 ' ∗,and since by definition ΩBM is the pullback of ∗ → BM ← ∗, we get acanonical map of E1-monoids M → ΩBM .

We are interested in the question of when M → ΩBM is an equivalence.In particular, this would display M as a loop space in a canonical way. Ofcourse, we cannot always expect M → ΩBM to be an equivalence sinceΩBM is, in classical terms, a H-group whereas M is only a H-space – i.e.we have not assumed that the E1-structure on M comes with inverses. TheE1-structure on M endows π0M with a canonical monoid structure, and atthis level our observation becomes that π0M is not always a group, whereasπ0(ΩBM) is always a group. If π0M is a group, we say that M is an E1-group.A theorem going back at least to [Sta63] states that the failure of π0M to bea group is the only obstruction to M → ΩBM being an equivalence:

28

2.1 Delooping E1-groups 2 GROUP COMPLETION

2.2 Theorem (Stasheff). Let M be an E1-monoid. The canonical map M →ΩBM is an equivalence if and only if M is an E1-group.

Proof. As discussed above, the “only if” statement is trivial. To prove thenontrivial “if” statement, we follow Segal’s proof from [Seg74]. First notethat π0M is a group if and only if the shear maps

M ×M (d0,d2)←−−−−∼

M2(d0,d1)−−−−→M ×M and M ×M (d0,d2)←−−−−

∼M2

(d2,d1)−−−−→M ×M

are equivalences. These are the maps M ×M → M ×M mapping (m,m′)to (mm′,m) and (m,mm′) respectively. Note that the first shear map is anequivalence if and only if the diagram

M2 M

M M0

d0

d1

d0

d1

is a pullback. The same observation holds for the second shear map, and itfollows more generally (using simplicial identities and the Segal condition)that

Mn+1 Mm+1

Mn Mm

d0

ψ∗

d0

θ∗

(2.1)

is a pullback where θ : [m]→ [n] is any map and ψ : [m+ 1]→ [n+ 1] is themap defined by ψ(0) = 0 and ψ(k) = ψ(k − 1) + 1.

Recall that the simplicial space A has a path-space

PA• = A• P : N∆op → S

where P : N∆ → N∆ is the functor taking [n] to [n + 1] and θ : [m] → [n]to the map ψ : [m + 1] → [n + 1] defined as in diagram (2.1). Moreover,the coface operator d0 defines a natural transformation id → P , and thus acanonical map PA• → A•. Note that by definition PAn = An+1. Crucially,‖PA•‖ ' ∗. To prove the theorem, it therefore suffices to show that

M1 PM0 ‖PM‖

M0 BM

29

2.2 The group completion theorem 2 GROUP COMPLETION

is a pullback. But according to the following lemma (specifically the n = 0case of the conclusion), this follows from the fact that the diagram (2.1) is apullback (for all θ).

2.3 Lemma ([Seg74, 1.6]). Let f : A′• → A• be a map of simplicial spaces sothat

A′n A′m

An Am

θ∗

fn fm

θ∗

is a pullback for each θ : [m]→ [n]. Then for each n, the diagram

A′n ‖A′•‖

An ‖A•‖

fn

is a pullback.

2.2 The group completion theorem

Let M be an E1-monoid, let R be an E∞-ring spectrum (e.g. HZ or S), andlet πR be the image of the Hurewicz map π0M → R0M .

2.4 Theorem (Group completion). Suppose that πR satisfies the leftOre condition in R∗M . Then the canonical map M → ΩBM induces anisomorphism of rings

π−1R R∗(M)

∼−→ R∗(ΩBM).

Proof. We claim that the functor ΩB : MonE1(S) → MonE1(S) is a localiza-tion with essential image GrpE1

(S). The canonical maps M → ΩBM formthe components of a natural transformation, so according to Proposition 1.15it suffices to check that both canonical maps ΩBM ⇒ ΩBΩBM are equiva-lences. But since ΩBM is grouplike, this follows from Theorem 2.2.

We now claim that R[M ] → R[ΩBM ] exhibits R[ΩBM ] as the localiza-tion of R[M ] at πR as an E1-algebra. Note that ordinarily R[−] is a functor

30

2.3 The (−)∞ construction 2 GROUP COMPLETION

from S to LModR, explicitly defined as R ⊗ Σ∞+ (−). At this level we haveadjunctions

LModR Sp SR⊗−

Ω∞

Σ∞+

All of these ∞-categories are symmetric monoidal. The functors R⊗− andΣ∞+ are monoidal, so their adjoints are lax monoidal by easy nonsense. Theupshot is that we can promote everything to the level of E1-algebras:

AlgE1(LModR) AlgE1

(Sp) MonE1(S)R⊗−

Ω∞

Σ∞+

The claim that R[M ] → R[ΩBM ] is a localization follows from the adjunc-tion. The theorem now follows from Theorem 1.33.

2.3 The (−)∞ construction

Assume from now on that M is an E∞-monoid. This implies that the Orecondition comes for free. More importantly, it permits the following con-struction:

2.5 Construction. Let X be a left M -space. Since M is an E∞-monoid,each multiplication X

m−→ X (m ∈M) is a map of left M -spaces. Explicitly,for each m,n ∈ M the E∞-structure on M produces a canonical homotopyh : n ·(m ·−) ' m ·(n ·−) which is determined by the path τ(m,n) : t 7→ h(t, e),where e denotes the unit of M . But then

X X

X X

n

m

nτ(m,n)

m

is a commutative diagram in the ∞-categorical sense, where the 2-cell ismultiplication by τ(m,n). Hence the colimit

Xm = lim−→(Xm−→ X

m−→ X → . . . )

may be taken in the category of left M -spaces, and is therefore again a leftM -space.

31


Once and for all, fix a well-ordered generating set T of M . For each finitesubset S ⊆ T , we may write S = m1, . . . ,mn such that m1 < · · · < mn.Inductively, define

XS = (Xm1,...,mn−1)mn .

Finally, putX∞ = lim−→

S

XS,

where the colimit is taken over finite subsets S ⊆ T .

Obviously the construction is well-defined and functorial. In particular,i : M → ΩBM induces M∞ → (ΩBM)∞. Since the action of M on ΩBMis by equivalences, there is a canonical equivalence (ΩBM)∞ ' ΩBM . Pre-composing with the canonical map M →M∞, we get a sequence of canonicalmaps:

M →M∞ → ΩBM.

This sequence is our next object of study. Note that M∞ → ΩBM is astable equivalence (i.e. induces isomorphisms on stable homotopy) by thegroup completion theorem. In general, however, this map will not be a trueequivalence. We will study an intersting counterexample in § 2.5. First,however, we want to know when the map is an equivalence.

2.6 Construction. For each n ≥ 2, the fact that M is an E∞-monoidimplies that the multiplication map Mn →M factors through the homotopycoinvariants (Mn)hΣn of the action which permutes coordinates.3

For each m ∈ M , let (m, . . . ,m) : BΣn → (Mn)hΣn be the map inducedby (m, . . . ,m) : ∗ →Mn. Then m defines a map

φ(m) : BΣn → (Mn)hΣn →M →M∞.

2.7 Proposition. The following are equivalent:

(1) The canonical action of π0M on M∞ is via equivalences.

(2) The map M → M∞ is a localization with respect to left M-spaces onwhich π0M acts via equivalences.

3Outside the world of higher nonsense, a roundabout way of saying that a monoid M(in Sets) is commutative is that the n-fold multiplication map Mn → M factors throughthe coinvariants (Mn)Σn

of the action which permutes coordinates.

32


(3) The canonical map M∞ → ΩBM is an equivalence.

(4) The fundamental groups of M∞ at all basepoints are abelian.

(5) The fundamental groups of M∞ at all basepoints are hypoabelian, mean-ing that they have no nontrivial perfect subgroups.

(6) For each m ∈ T , the map

φ(m)∗ : Σ3 → π1(M,m3)→ π1(M∞,m3)

sends (123) to zero.

(7) For each m ∈ T there is some n ≥ 2 such that

φ(m)∗ : Σn −→ π1(M,mn)→ π1(M∞,mn)

sends (12 . . . n) to zero.

Proof of Proposition 2.7. Note that (3) ⇒ (4), (4) ⇒ (5) and (6) ⇒ (7) aretrivial. Furthermore, (4)⇒ (6) follows from the algebraic fact that (123) liesin the commutator subgroup of Σ3, and similarly (5) ⇒ (7) follows from thefact that (12 . . . n) lies in the largest perfect subgroup of Σn for odd n ≥ 5.It now suffices to show (1) ⇒ (2), (2) ⇒ (3) and (7) ⇒ (1).

Of the remaining implications, we first prove (1) ⇒ (2). Say that a leftM -space X is π0M-local if π0M acts invertibly on X. By assumption M∞ isπ0M -local, so it suffices to show that the induced map on mapping spaces

Map(M∞, X)→ Map(M,X)

is an equivalence for every π0M -local space X. (Here the mapping spacesare taken in the ∞-category of left M -spaces LModM(S), but we suppressthis in our notation.) Due to the recursive definition M∞, we may assumethat the set of generators T from earlier consists of a single point m ∈ M .But then

Map(M∞, X) = Map(lim−→(Mm−→M

m−→ . . . ), X)

' lim←−(

Map(M,X)m∗−→ Map(M,X)

m∗−→ . . .)

' lim←−(

Map(M,X)m∗−→ Map(M,X)

m∗−→ . . .).

33


In the last expression, note that m∗ is an equivalence since X is π0M -local.Hence the limit identifies with Map(M,X) and this identification comes fromthe canonical map M →M∞.

To show that (2) ⇒ (3), it suffices to show that the localization of M(at π0M) as a left M -space is automatically its localization (at π0M) as anE1-monoid, and hence coincides with the group completion of M . But thisfollows from the same nonsense as in the proof of Corollary 1.25.

It remains to be seen that (7) ⇒ (1). Again we assume the generatingset T of M consists of a single generator m. To show (1), it clearly sufficesto show that m acts invertible on M∞. In other words, we must show thatthe map induced on colimits by

M M M . . .

M M M . . .

m

m

m

m

τm

m

τ

m m m

(2.2)

is an equivalence. Here τ denotes the symmetry path τ(m,m) from earlier,which is a loop here. For once we have included the diagonals and 2-cells,since these will be relevant to our argument. There are no issues with highercoherence since the indexing category is (N,≤).

We claim that the map induced by

M M M . . .

M M M . . .

id

m

id

m

idid

m

id

m m m

is an inverse to the map induced by (2.2). In both directions, the compositionis induced by

M M M . . .

M M M . . .

m

m

τ

m

mτ

m

m

m m m

(2.3)

Note that the loop τ corresponds to the image of the transposition (12) underΣ2 → π1(M,m2). More generally, the composition of n − 1 consecutive 2-cells in diagram (2.3) corresponds to the image of (12 . . . n) under Σn →π1(M,mn). Our assumption says that this 2-cell is equivalent to the identity2-cell for some n ≥ 2. It follows that the map coming from diagram (2.3) isthe identity.

34

2.4 Connection with the plus construction 2 GROUP COMPLETION

2.8 Corollary. Suppose that M has hypoabelian fundamental groups at everybasepoint. Then M∞ → ΩBM is an equivalence, so in particular M∞ hasabelian fundamental groups.

Proof. For each m ∈ T and n ≥ 5, the first map in the composition Σn →π1(M,mn) → π1(M∞,m

n) must annihilate (12 . . . n), since the latter liesin the largest perfect subgroup of Σn. The conclusion now follows fromTheorem 2.7(7).

2.4 Connection with the plus construction

Classically (see [HH79]), the plus construction on a space X is a map i : X →X+ such that

• The map i is acyclic, meaning that it induces isomorphisms on homol-ogy in all local coefficient systems.

• For each basepoint x0 ∈ X, the induced map π1(X, x0)→ π1(X+, i(x0))is surjective and its kernel is the largest perfect subgroup of π1(X, x0).

The second condition implies in particular that X+ has hypoabelian funda-mental groups at all basepoints.

Since local coefficient systems completely detect obstructions to liftingproblems (see [Whi12]), this definition characterizes X → X+ as the unitof an adjunction. Here the right adjoint is the inclusion Shypo ⊆ S of thefull subcategory spanned by spaces whose fundamental groups are hypoa-belian at all basepoints and the left adjoint is the plus construction functor(−)+. When viewed as a functor from spaces to spaces, the plus constructionpreserves products, so in particular it preserves E∞-monoids.

The following theorem is hard-won in the classical approach to groupcompletion, but follows easily from our setup:

2.9 Theorem (Randal-Williams [RW13], Nikolaus [Nik17]). Both canon-ical maps

(M+)∞ ←M∞ → ΩBM

35

2.5 The Barratt-Priddy-Quillen theorem 2 GROUP COMPLETION

are plus constructions. In particular,

(M+)∞ ' (M∞)+ ' ΩBM.

Proof. We start by checking that M∞ → (M+)∞ is a plus construction. Firstnote that (M+)∞ has hypoabelian (even abelian!) fundamental groups byCorollary 2.8. Hence all that remains is to prove that for each space X withhypoabelian fundamental groups, the induced map

MapS((M+)∞, X)→ MapS(M∞, X)

is an equivalence. For simplicity, we again assume that the generating set Tconsists of a single generator. But then

MapS((M+)∞, X) = MapS(lim−→(M+ m−→M+ m−→ . . . ), X)

' lim←−(

MapS(M+, X)

m∗−→ MapS(M+, X)

m∗−→ . . .)

' lim←−(

MapS(M,X)m∗−→ MapS(M,X)

m∗−→ . . .)

' MapS(M∞, X),

proving the claim.We now show that M∞ → ΩBM is a plus construction. Functoriality

gives a commutative diagram

(M∞)+ ΩBM

(M+)∞ ΩB(M+)

∼ ∼

∼

Here the lefthand vertical arrow is an equivalence by what we have proved;the bottom arrow is an equivalence by Corollary 2.8; and the righthandvertical arrow is a stable equivalence by the group completion theorem, andthus an equivalence since both spaces are loop spaces.

2.5 The Barratt-Priddy-Quillen theorem

In this subsection, we apply the theory from above to show that each path-component of the stable sphere Ω∞S is equivalent to BΣ+

∞, the plus con-struction on the classifying space of the group of permutations of Z≥0 which

36


fix all but finitely many elements. In order to arrive at this beautiful andsurprising theorem, we must first recall the Barratt-Priddy-Quillen theorem.

Let SymMonCat (resp. SymMonGrpd) denote the category of small sym-metric monoidal categories (resp. groupoids) and monoidal functors. Theforgetful functor

SymMonGrpd→ SymMonCat

has a right adjoint (−)∼. Explicitly, if C is a small symmetric monoidalcategory, we define C∼ to be the monoidal subcategory of C containing allobjects of C, but having as morphisms only the isomorphisms of the latter.The counit of this adjunction is the inclusion C∼ ⊆ C.

Consider the sequence of adjunctions

Sets SymMonCat SymMonGrpd MonE∞(S) GrpE∞(S),U

U

(−)∼

h

N

ΩB

U

where the functors labelled U are forgetful.

2.10 Definition. Algebraic K-theory is the composite functor

K = ΩBN(−)∼ : SymMonCat→ GrpE∞(S).

2.11 Theorem (Barratt-Priddy-Quillen). There is an equivalence of E∞-groups

Ω∞S ' K(Fin),

where Fin denotes the skeletal category of finite sets with the symmetricmonoidal structure of disjoint union.

Sketch of proof. The forgetful functor MonE∞(S) → S has a left adjoint Fwhich at the level of objects is given by

F : X 7→∐n≥0

(Xn)hΣn ,

where Σn acts on Xn by coordinate permutation. Note that

F (∗) '∐n≥0

BΣn.

37


Since ΩB is left adjoint to the forgetful functor GrpE∞(S) → MonE∞ , thecomposite functor ΩB F : GrpE∞(S) → S is left adjoint to the forgetfulfunctor from E∞-groups to spaces. Hence

ΩBF (∗) ' ΩB∐n≥0

BΣn (2.4)

is the free E∞-group on a single generator.There is another way to compute the free E∞-group on a single generator.

A classical fact states that connective spectra are the same as E∞-groups(see [Seg74, 3.4]). More precisely,

Ω∞ : Spcn ∼−→ GrpE∞(S).

is an equivalence (HA 5.2.6.26). From the commutative diagram

Spcn GrpE∞(S)

S

Ω∞

∼

Σ∞+

Ω∞ ΩBF

we see thatΩBF (∗) = Ω∞Σ∞+ (∗) = Ω∞S. (2.5)

Combining (2.4) and (2.5), we find that Ω∞S ' ΩB∐

n≥0BΣn. But bydefinition K(Fin) = ΩBN(Fin∼) = ΩB

∐n≥0BΣn.

2.12 Corollary. There is an equivalence of spaces

Z×BΣ+∞ ' Ω∞S.

Proof. The E1-monoid M =∐

n≥0BΣn is generated by the basepoint m0

in BΣ1. As may be seen at the symmetric monoidal groupoid level, thisbasepoint acts via shifting M , meaning that multiplication by m0 embedsBΣn in BΣn+1. The particular embedding BΣn → BΣn+1 is induced bythe injection Σn → Σn+1 which sends an n-permutation σ to the (n + 1)-permutation which acts by σ on the first n elements while fixing the lastelement. Hence M∞ ' Z × BΣ∞, where the n-th summand is contributed

38

A APPENDIX

by the diagonal starting at the cell labelled n in the following picture:

∞∐

n≥0BΣn

∐n≥0BΣn

∐n≥0BΣn

∐n≥0BΣn · · ·

......

......

...

2 BΣ2 BΣ2 BΣ2 BΣ2 · · ·

1 BΣ1 BΣ1 BΣ1 BΣ1 · · ·

0 BΣ0 BΣ0 BΣ0 BΣ0 · · ·

0 −1 −2 −3 · · ·

The conclusion now follows from Theorem 2.9.

A Appendix

Let Fin∗ denote the category of finite pointed sets and pointed maps. Foreach n ≥ 0, let 〈n〉 denote the set 0, . . . , n with 0 as its basepoint; also, foreach 1 ≤ i ≤ n let ρi : 〈n〉 → 〈1〉 denote the morphism in Fin∗ that sends ito 1 and everything else to 0.

A.1 Definition. An E∞-monoid is a functor M : N(Fin∗)→ S such that

(1) The space M〈0〉 is contractible.

(2) The induced maps (ρi)∗ : M〈n〉 → M〈1〉 exhibit M〈n〉 as the n-foldproduct of M〈1〉.

A.2 Definition. The cut functor is the faithful contravariant functor cut : ∆→Fin∗ given

• on objects by sending [n] to 〈n〉;

• on morphisms by sending θ : [m] → [n] to the map f : 〈n〉 → 〈m〉determined by

f−1(0) = i ≤ θ(0) ∪ θ(m) < i ≤ nf−1(k) = θ(j − 1) < i ≤ θ(j)

for 0 < k ≤ n.

39

A APPENDIX

The cut functor allows us to think of ∆op as a subcategory of Fin∗ withthe same objects but far fewer morphisms. However, the morphisms ρi whichappear in Definition A.1 are contained in (the image of) ∆op. Explicitly,ρi = cut(ρi) where ρi : [1]→ [n] is the morphism that sends 0 to i− 1 and 1to i.

A.3 Definition. An E1-monoid is a functor M : N∆op → S such that

(1) The space M [0] is contractible.

(2) The induced maps (ρ1)∗ : M [n] → M [1] exhibit M [n] as the n-foldproduct of M [1].

More generally, a functor A : N∆op → S is just a (homotopy coherent)simplicial space. To emphasize this, we write A• = A and An = A[n].

Let M• be an E1-monoid. Intuitively, the fact that ρ1, . . . , ρn exhibit Mn

as the n-fold product of M1 suggests that we should think of M• as a kind ofbar construction on M1. That is, forgetting higher coherence we can pictureM• as a diagram

· · · Mn · · · M2 M1 M0 ' ∗...

......

where the indicated arrows are face and degeneracy operators. Thus the mid-dle arrow d1 : M2 →M1 should be though of as multiplication. Functorialityof M• encodes the homotopy coherent associativity of this multiplication.

Similarly, choosing an equivalence M0 ' ∗ equips M1 with a basepointvia the degeneracy operator M0 → M1, and simplicial identities show thatthis basepoint behaves like a homotopy coherent unit with respect to themultiplication map.

We will think of M1 as the underlying monoid of M• and write M = M1.An E∞-monoid can be forgetfully viewed as an E1-monoid by precom-

posing with the cut functor. The additional structure on E∞-monoids comesfrom the fact that there are more maps in Fin∗. Concretely, the extra mapsare those which are not order-preserving. For instance, there is a switchingmorphism 〈2〉 → 〈2〉:

2 2

1 1

0 0

40

A APPENDIX

Composing with the multiplication morphism, we find that

2 2

1 1 1

0 0 0

=

2

1 1

0 0

By functoriality, when we pass to spaces we therefore have a commutativediagram in the ∞-categorical sense:

M2 M

M2

mult.

switch mult.

Along with the remaining structure, this encodes homotopy coherent com-mutativity of the multiplication in an E∞-monoid.

41

REFERENCES REFERENCES

References

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https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/

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Group completion is a completion

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